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Distinguishing graphs by their spectra, Smith normal forms and complements

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  • Abiad, Aida
  • Alfaro, Carlos A.
  • Villagrán, Ralihe R.

Abstract

The search for a highly discriminating and easily computable invariant to distinguish graphs remains a challenging research topic. Here we focus on cospectral graphs whose complements are also cospectral (generalized cospectral), and on coinvariant graphs (same Smith normal form) whose complements are also coinvariant (generalized coinvariant). We show a new characterization of generalized cospectral graphs in terms of codeterminantal graphs. We also establish the Smith normal form of some graph classes for certain associated matrices, and as an application, we prove that the Smith normal form can be used to uniquely determine star graphs. Finally, for graphs up to 10 vertices, we present enumeration results on the number of generalized cospectral graphs and generalized coinvariant graphs with respect to several associated matrices.

Suggested Citation

  • Abiad, Aida & Alfaro, Carlos A. & Villagrán, Ralihe R., 2025. "Distinguishing graphs by their spectra, Smith normal forms and complements," Applied Mathematics and Computation, Elsevier, vol. 490(C).
    • Handle: RePEc:eee:apmaco:v:490:y:2025:i:c:s0096300324006593
    DOI: 10.1016/j.amc.2024.129198
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    References listed on IDEAS

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    1. Alfaro, Carlos A. & Villagrán, Ralihe R., 2021. "The structure of sandpile groups of outerplanar graphs," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    2. van Dam, E.R. & Haemers, W.H. & Koolen, J.H., 2006. "Cospectral Graphs and the Generalized Adjacency Matrix," Discussion Paper 2006-31, Tilburg University, Center for Economic Research.
    3. van Dam, E.R. & Haemers, W.H., 2002. "Which Graphs are Determined by their Spectrum?," Discussion Paper 2002-66, Tilburg University, Center for Economic Research.
    4. Rakshith, B.R. & Das, Kinkar Chandra, 2023. "On distance Laplacian spectral determination of complete multipartite graphs," Applied Mathematics and Computation, Elsevier, vol. 443(C).
    5. Aouchiche, Mustapha & Hansen, Pierre, 2018. "Cospectrality of graphs with respect to distance matrices," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 309-321.
    6. Abiad, Aida & Alfaro, Carlos A., 2021. "Enumeration of cospectral and coinvariant graphs," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    Full references (including those not matched with items on IDEAS)

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